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Multivariate Description. What Technique?. Raw Data. Linear Regression. Two Regressions. Principal Components. Gulls Variables. Scree Plot. Output. > summary(gulls.pca2) Importance of components:

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Multivariate Description

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Multivariate description

Multivariate Description

What technique

What Technique?

Raw data

Raw Data

Linear regression

Linear Regression

Two regressions

Two Regressions

Principal components

Principal Components

Gulls variables

Gulls Variables

Scree plot

Scree Plot



> summary(gulls.pca2)

Importance of components:

Comp.1 Comp.2 Comp.3 Standard deviation 1.8133342 0.52544623 0.47501980 Proportion of Variance 0.8243224 0.06921464 0.05656722 Cumulative Proportion 0.8243224 0.89353703 0.95010425

> gulls.pca2$loadings


Comp.1 Comp.2 Comp.3 Comp.4Weight -0.505 -0.343 0.285 0.739Wing -0.490 0.852 -0.143 0.116Bill -0.500 -0.381 -0.742 -0.232H.and.B -0.505 -0.107 0.589 -0.622

Bi plot


Environmental gradients

Environmental Gradients

Models of species response

Models of Species Response

There are (at least) two models:-

  • Linear - species increase or decrease along the environmental gradient

  • Unimodal - species rise to a peak somewhere along the environmental gradient and then fall again





Inferring gradients from attribute data e g species

Inferring Gradients from Attribute Data (e.g. species)

Indirect gradient analysis

Indirect Gradient Analysis

  • Environmental gradients are inferred from species data alone

  • Three methods:

    • Principal Component Analysis - linear model

    • Correspondence Analysis - unimodal model

    • Detrended CA - modified unimodal model

Terschelling dune data

Terschelling Dune Data

Pca gradient site plot

PCA gradient - site plot

Pca gradient site species biplot

PCA gradient - site/species biplot


biodynamic& hobby


Making effective use of environmental variables

Making Effective Use of Environmental Variables



  • Use single responses in linear models of environmental variables

  • Use axes of a multivariate dimension reduction technique as responses in linear models of environmental variables

  • Constrain the multivariate dimension reduction into the factor space defined by the environmental variables

Ordination constrained by the environmental variables

Ordination Constrained by the Environmental Variables



Working with the variability that we can explain

Working with the Variability that we Can Explain

  • Start with all the variability in the response variables.

  • Replace the original observations with their fitted values from a model employing the environmental variables as explanatory variables (discarding the residual variability).

  • Carry our gradient analysis on the fitted values.

Unconstrained constrained


  • Unconstrained ordination axes correspond to the directions of the greatest variability within the data set.

  • Constrained ordination axes correspond to the directions of the greatest variability of the data set that can be explained by the environmental variables.

Direct gradient analysis

Direct Gradient Analysis

  • Environmental gradients are constructed from the relationship between species environmental variables

  • Three methods:

    • Redundancy Analysis - linear model

    • Canonical (or Constrained) Correspondence Analysis - unimodal model

    • Detrended CCA - modified unimodal model

Dune data unconstrained

Dune Data Unconstrained

Dune data constrained

Dune Data Constrained

Similarity approaches

Similarity approaches

Different types of data

Different types of data


Continuous data:height

Categorical data

ordered (nominal):growth rate

very slow, slow, medium, fast, very fast

not ordered:fruit colour

yellow, green, purple, red, orange

Binary data:fruit / no fruit

Similarity matrix

Similarity matrix

We define a similarity between units – like the correlation between continuous variables.

(also can be a dissimilarity or distance matrix)

A similarity can be constructed as an average of the similarities between the units on each variable.

(can use weighted average)

This provides a way of combining different types of variables.

Distance metrics





Distance metrics

relevant for continuous variables:


city block or Manhattan

(also many other variations)

Similarity coefficients for binary data









Similarity coefficients for binary data

simple matching

count if both units 0 or both units 1


count only if both units 1

(also many other variants, eg Bray-Curtis)

simple matching can be extended to categorical data

A distance matrix

A Distance Matrix

Uses of distances

Uses of Distances

Distance/Dissimilarity can be used to:-

  • Explore dimensionality in data using Principal coordinate analysis (PCO or PCoA)

  • As a basis for clustering/classification

Uk wet deposition network

UK Wet Deposition Network

Non metric multidimensional scaling

Non-metric multidimensional scaling

NMDS maps the observed dissimilarities onto an ordination space by trying to preserve their rank order in a low number of dimensions (often 2) – but the solution is linked to the number of dimensions chosen

it is like a non-linear version of PCO

define a stress function and look for the mapping with minimum stress

(e.g. sum of squared residuals in a monotonic regression of NMDS space distances between original and mapped dissimilarities)

need to use an iterative process, so try with many different starting points and convergence is not guaranteed

Procrustes rotation

Procrustes rotation

used to compare graphically two separate ordinations

Grouping methods

Grouping methods

Cluster analysis

Cluster Analysis

Clustering methods

Clustering methods

  • hierarchical

    • divisive

      • put everything together and split

      • monothetic / polythetic

    • agglomerative

      • keep everything separate and join the most similar points (classical cluster analysis)

  • non-hierarchical

    • k-means clustering

Agglomerative hierarchical

Agglomerative hierarchical

Single linkage or nearest neighbour

finds the minimum spanning tree:

shortest tree that connects all points

  • chaining can be a problem

Agglomerative hierarchical1

Agglomerative hierarchical

Complete linkage or furthest neighbour

  • compact clusters of approximately equal size.

  • (makes compact groups even when none exist)

Agglomerative hierarchical2

Agglomerative hierarchical

Average linkage methods

  • between single and complete linkage

From alexandria to suez

From Alexandria to Suez

Hierarchical clustering

Hierarchical Clustering

Hierarchical clustering1

Hierarchical Clustering

Hierarchical clustering2

Hierarchical Clustering

Building and testing models

Building and testing models

Basically you just approach this in the same way as for multiple regression – so there are the same issues of variable selection, interactions between variables, etc.

However the basis of any statistical tests using distributional assumptions are more problematic, so there is much greater use of randomisation tests and permutation procedures to evaluate the statistical significance of results.

Some examples

Some examples

Multivariate description

Part of Fig 4.

Multivariate description

Positive Matrix Factorisation


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